Sen. Chuck Grassley, R-Iowa, the ranking Republican on the Senate Judiciary Committee makes a point on Capitol Hill in Washington, on Monday as lawmakers work on a landmark immigration bill to secure the border and offer citizenship to millions.

WASHINGTON — Senate supporters of far-reaching immigration legislation accepted minor changes in public while negotiating over more sweeping alterations in private Monday as they drove toward expected Judiciary Committee approval by mid-week.

The legislation would provide an opportunity of U.S. citizenship to millions of immigrants living in the country illegally, create a new visa program for low-skilled workers and permit a sizeable increase in the number of high-tech visas, at the same time it mandates new measures to crack down on future unlawful immigration.

In two previous weeks of deliberations, supporters of the legislation have demonstrated their command over the committee’s proceedings, alternately accepting some proposals advanced by the bills critics and rejecting others — all without losing a single showdown.

The same pattern held true as the committee embarked on its third and final week of drafting.

On a vote of 13-5, the legislation’s supporters agreed to require foreigners leaving the country through any of the nation’s 30 busiest airports to submit to fingerprinting, part of an attempt to strengthen security. The proposal was drafted by Sen. Orrin Hatch, R-Utah, who has yet to announce whether he will support the bill.

“This is an agreement that we need to build toward a biometric visa exit system,” said Sen. Jeff Flake, R-Ariz. He said the system is “long overdue.” Most of the committee’s Democrats supported the provision, along with four Republicans. Among them were GOP Sens. Lindsey Graham of South Carolina and Jeff Flake of Arizona, two the so-called Gang of Eight that negotiated the bill’s basic framework over many months.

The committee last week rejected a proposal by Sen. Jeff Sessions, R-Ala., to impose the fingerprinting requirement at all of the nation’s airports rather than only the biggest. He said the system’s partial implementation marked a “retreat from current law,” which already requires a nationwide biometric system. The requirement has not been fulfilled because of the cost.

While the committee met, officials said a complex private negotiation was playing out over proposed changes to a section of the legislation that would expand the current cap on H-1B high skilled visas from 65,000 annually to 110,000, with the possibility of a further rise to 180,000.

Lawmakers, aides and lobbyists familiar with the talks said that Hatch, whose state of Utah is home to a burgeoning high tech industry, was seeking to ease the terms and costs that would apply to companies that make use of highly skilled immigrant workers.

In general, organized labor and its allies on the committee, including Sen. Richard Durbin, D-Ill., wanted tougher conditions than industry was seeking, part of an attempt to assure than American workers are not disadvantaged by a larger influx of H1-B visa holders.

In an email issued during the afternoon, the AFL-CIO urged its supporters to contact the committee and express opposition to “Hatch’s anti-worker amendments...

“Hatch’s amendments would change the bill so high-tech companies can hire new immigrant employees without first making the jobs available to American workers,” it said. “Hatch’s amendments would mean American corporations could fire American workers in order to bring in new immigrant workers at lower wages.”

A short while later, Hatch’s spokesman, Matthew Harakal, said the Utah Republican is “encouraged with where discussions are headed and hopes that an agreement can be reached on something he can support.”

One Senate official said two earlier stabs at compromise had failed, one when the high tech industry rebelled at the terms, and the other when the AFL-CIO balked. The officials who described the deliberations did so on condition of anonymity, saying they were not authorized to speak on the record.

The bill’s supporters are eager to strike a deal with Hatch, although officials say they would insist that he agree to vote for the legislation in committee if they made concessions he wanted.

Whatever the ultimate outcome of the negotiations, it seemed unlikely that the committee’s vote would be the final vote on high tech visas — or any other major portion of the measure.

“This bill is not perfect,” Schumer said at one point during the day, one of several occasions in the past two weeks he has signaled that he and other supporters will be receptive to some changes when the legislation reaches the Senate floor.

And in fact, Sen. Marco Rubio, R-Fla., arguably the Republican with the largest ability to sway the vote in the Senate, has said he wants to tighten the fingerprinting requirement that the committee accepted last week. “I will continue to fight to make the tracking of entries and exits include biometrics in the most effective system we can build when the bill is amended on the Senate floor,” he said.

In other votes, the committee approved more visas for Tibetans and increased information sharing among federal agencies when people overstay their visas.

It also voted to strip asylum or refugee status from individuals who return to the country they fled, unless they can show a good reason for doing so.

Graham, R-S.C., said the asylum change was merited in light of the Boston Marathon bombings. The brothers who allegedly set off the bombs arrived in the U.S. as boys when their family sought asylum here.